I think most people people would consider "fat" offensive. "thin" can be used in a positive or negative manner. Some women might admire another woman for being "thin". As soon as you say "too thin" or so, it's negative of course. "anorexic" is one way of saying thin of saying who looks thin in an unhealthy way, however I think it's not a good word to use because it immediately makes a connection with anorexia, an eating disorder. I think people should leave medical diagnoses to the medical profession. Not everyone who someone at a glance might consider "too thin" is "too thin" in medical terms, and even if they are, their low weight may be caused by something wholly else than anorexia. If something looks scarily thin (i.e. you're concerned about this person's health) then I'd just say precisely that: scarily thin. Here's a nice write-up of different general adjectives you can use: http://www.differencebetween.com/difference-between-thin-and-vs-skinny-and-vs-slim-and-vs-slender-and-vs-lean/

I agree with the previous statements. Referring to someone's weight is usually offensive no matter how you say it if that person is overweight, but this only applies if you are talking directly to that person. If you are talking to someone and you call somebody else fat it might be considered as insensitive, especially if you are just talking with an acquaintance. Among friends it is much more common, but it is still considered rude. Overweight is the neutral term. It doesn't express any negative judgment really. Plump is more of a humorous term.

I agree with the answer above. But to answer your other questions: Any of the overweight terms can be considered offensive by some people if they are unhappy with their weight so use them with caution. But you do have them in the correct order of politeness. Your thoughts on thin are also in the correct order but most Americans would not be offended by being called any of them.